Landsberg-Schaar relationLandsbergSchaarRelation2003-01-25 00:30:162006-07-24 20:21:45Theorem\usepackage{amssymb}
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The Landsberg-Schaar relation states that for any positive integers $p$
and $q$: \begin{equation}
\frac{1}{\sqrt{p}}\sum_{n=0}^{p-1}\exp\left(\frac{2\pi in^2q}{p}\right)=
\frac{e^{\pi i/4}}{\sqrt{2q}}\sum_{n=0}^{2q-1}\exp\left(-\frac{\pi in^2p}{2q}\right)
\end{equation}
Although both sides of (1) are mere finite sums,
no one has yet found a proof which uses no infinite
limiting process. One way to prove it is to put
$\tau=2iq/p+\epsilon$, where $\epsilon>0$, in
this identity due to Jacobi:
\begin{equation}
\sum_{n=-\infty}^{+\infty}e^{-\pi n^2\tau}=\frac{1}{\sqrt{\tau}}
\sum_{n=-\infty}^{+\infty}e^{-\pi n^2/\tau}
\end{equation}
\noindent
and let $\epsilon\to 0$. The details can be found \PMlinkname{here}{ProofOfJacobisIdentityForVarthetaFunctions}. The identity (2) is a basic one in the theory of
theta functions. It is sometimes called the functional equation for the Riemann theta function. See e.g. [2 VII.6.2].
If we just let $q=1$ in the Landsberg-Schaar identity, it reduces to a formula
for the quadratic Gauss sum mod $p$; notice that $p$ need not be prime.
\textbf{References:}
\noindent
[1] H. Dym and H.P. McKean. \emph{Fourier Series and Integrals}. Academic Press, 1972.
\noindent
[2] J.-P. Serre. \emph{A Course in Arithmetic}. Springer, 1970.